Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains

In this paper we study the $L^p$ boundary value problems for $\mathcal{L}(u)=0$ in ${\mathbb{R}}_{+}^{d+1}$, where $\mathcal{L}=-\text{div}(A\mathrm{\nabla })$ is a second order elliptic operator with real and symmetric coefficients. Assume that $A$ is {\it periodic} in $x_{d+1}$ and satisfies some minimal smoothness condition in the $x_{d+1}$ variable, we show that the $L^p$ Neumann and regularity problems are uniquely solvable for $1<p<2+\delta$. We also present a new proof of Dahlberg’s theorem on the $L^p$ Dirichlet problem for $2-\delta <p<\mathrm{\infty }$ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the $x_{d+1}$ variable, these results extend directly from ${\mathbb{R}}_{+}^{d+1}$ to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform $L^p$ estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.